Browse > Article
http://dx.doi.org/10.12989/sem.2011.40.5.705

Parametrically excited viscoelastic beam-spring systems: nonlinear dynamics and stability  

Ghayesh, Mergen H. (Department of Mechanical Engineering, McGill University)
Publication Information
Structural Engineering and Mechanics / v.40, no.5, 2011 , pp. 705-718 More about this Journal
Abstract
The aim of the investigation described in this paper is to study the nonlinear parametric vibrations and stability of a simply-supported viscoelastic beam with an intra-span spring. Taking into account a time-dependent tension inside the beam as the main source of parametric excitations, as well as employing a two-parameter rheological model, the equations of motion are derived using Newton's second law of motion. These equations are then solved via a perturbation technique which yields approximate analytical expressions for the frequency-response curves. Regarding the main parametric resonance case, the local stability of limit cycles is analyzed. Moreover, some numerical examples are provided in the last section.
Keywords
parametric vibrations; stability; viscoelastic materials; perturbation techniques;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
Times Cited By Web Of Science : 4  (Related Records In Web of Science)
Times Cited By SCOPUS : 4
연도 인용수 순위
1 Birman, V. (1986), "On the effects of nonlinear elastic foundation on free vibration of beams", ASME J. Appl. Mech., 53, 471-473.   DOI
2 Chen, L.Q. and Yang, X.D. (2005a), "Steady state response of axially moving viscoelastic beams with pulsating speed: comparison of two nonlinear models", Int. J. Solids. Struct., 42, 37-50.   DOI   ScienceOn
3 Chen, L.Q. and Yang, X.D. (2006), "Vibration and stability of an axially moving viscoelastic beam with hybrid supports", Eur. J. Mech., 25, 996-1008.   DOI   ScienceOn
4 Chen, L.Q., Zhang, N.H. and Zu, J.W. (2002), "Bifurcation and chaos of an axially moving visco-elastic strings", Chaos Soliton. Fract., 29, 81-90.
5 Chen, L.Q. and Yang, X.D. (2005b), "Stability in parametric resonance of axially moving viscoelastic beams with time-dependent speed", J. Sound. Vib., 284, 879-891.   DOI   ScienceOn
6 Chen, L.Q., Tang, Y.Q. and Lim, C.W. (2010), "Dynamic stability in parametric resonance of axially accelerating viscoelastic Timoshenko beams", J. Sound. Vib., 329, 547-565   DOI   ScienceOn
7 Cohen, Y.B. (2001), Electro Active Polymer (EPA) Actuators as Artificial Muscles, Reality, Potential, and Challenges, SPIE Press.
8 Darabi, M.A., Kazemirad, S. and Ghayesh, M.H. (2011), "Free vibrations of beam-mass-spring systems: Analytical analysis with numerical confirmation", Acta Mecha. Sinica. (in press)
9 Dowell, E.H. (1980), "Component mode analysis of nonlinear and nonconservative systems", ASME J. Appl. Mech., 47, 172-176.   DOI
10 Eisley, J.G. (1964), "Nonlinear vibration of beams and rectangular plates", ZAMP, 15, 167-175.   DOI   ScienceOn
11 Ghayesh, M.H. (2008), "Nonlinear transversal vibration and stability of an axially moving viscoelastic string supported by a partial viscoelastic guide", J. Sound. Vib., 314, 757-774.   DOI
12 Ghayesh, M.H. (2009), "Stability characteristics of an axially accelerating string supported by an elastic foundation", Mech. Machine Theory, 44, 1964-1979.   DOI   ScienceOn
13 Ghayesh, M.H. (2010), "Parametric vibrations and stability of an axially accelerating string guided by a nonlinear elastic foundation", Int. J. Nonlin. Mech., 45, 382-394.   DOI   ScienceOn
14 Ghayesh, M.H. and Balar, S. (2008), "Non-linear parametric vibration and stability of axially moving viscoelastic Rayleigh beams", Int. J. Solids. Struct., 45, 6451-6467.   DOI   ScienceOn
15 Ghayesh, M.H. and Balar, S. (2010), "Non-linear parametric vibration and stability analysis for two dynamic models of axially moving Timoshenko beams", Appl. Math. Model., 34, 2850-2859.   DOI   ScienceOn
16 Ghayesh, M.H. and Paidoussis, M.P. (2010a), "Dynamics of a fluid-conveying cantilevered pipe with intermediate spring support", ASME Conference Proceedings, 2010(54518), 893-902.
17 Ghayesh, M.H., Yourdkhani, M., Balar, S. and Reid, T. (2010), "Vibrations and stability of axially traveling laminated beams", Appl. Math. Comput., 217, 545-556.   DOI   ScienceOn
18 Ghayesh, M.H. and Moradian, N. (2011), "Nonlinear dynamic response of axially moving, stretched viscoelastic strings", Arch. Appl. Mech., 81(6), 781-799.   DOI   ScienceOn
19 Ghayesh, M.H. (2011), "On the natural frequencies, complex mode functions, and critical speeds of axially traveling laminated beams: Parametric study", Acta Mecha. Solida Sinica. (in press)
20 Ghayesh, M.H. and Paidoussis, M.P. (2010b), "Three-dimensional dynamics of a cantilevered pipe conveying fluid, additionally supported by an intermediate spring array", Int. J. Nonlin. Mech., 45(5), 507-524.   DOI   ScienceOn
21 Ghayesh, M.H., Paidoussis, M.P. and Modarres-Sadeghi, Y. (2011a), "Three-dimensional dynamics of a fluidconveying cantilevered pipe fitted with an additional spring-support and an end-mass", J. Sound. Vib., 330(12), 2869-2899.   DOI   ScienceOn
22 Ghayesh, M.H., Kazemirad, S., Darabi, M.A. and Woo, P. (2011b), "Thermo-mechanical nonlinear vibration analysis of a spring-mass-beam system", Arch. Appl. Mech. (in press)
23 Ghayesh, M.H., Alijani, F. and Darabi, M.A. (2011c), "An analytical solution for nonlinear dynamics of a viscoelastic beam-heavy mass system", J. Mech. Sci. Tech., 25(8), 1915-1923   DOI
24 Ghayesh, M.H., Kazemirad, S. and Darabi, M.A. (2011d), "A general solution procedure for vibrations of systems with cubic nonlinearities and nonlinear/time-dependent internal boundary conditions", J. Sound. Vib., 330(22), 5382-5400.   DOI   ScienceOn
25 Hu, K. and Kirmser, P.G. (1971), "On the nonlinear vibrations of free-free beams", ASME J. Appl. Mech., 38, 461-466.   DOI
26 Marynowski, K. (2006), "Two-dimensional rheological element in modelling of axially moving viscoelastic web", Eur. J. Mech. A-Soild., 25, 729-744.   DOI   ScienceOn
27 Karlik, B., Ozkaya, E., Aydin, S. and Pakdemirli, M. (1998), "Vibrations of a beam-mass systems using artificial neural networks", Comput. Struct., 69, 339-347.   DOI   ScienceOn
28 Marynowski, K. and Kapitaniak, T. (2002), "Kelvin-voigt versus burgers internal damping in modeling of axially moving viscoelastic web", Int. J. Nonlin. Mech., 37, 1147-1161.   DOI   ScienceOn
29 Marynowski, K. (2004), "Non-linear vibrations of an axially moving viscoelastic web with time-dependent tension", Chaos Soliton. Fract., 21, 2004, 481-490.   DOI   ScienceOn
30 Marynowski, K. and Kapitaniak, T. (2007), "Zener internal damping in modeling of axially moving viscoelastic beam with time-dependent tension", Int. J. Nonlin. Mech., 42, 118-131.   DOI   ScienceOn
31 Marynowski, K. (2010), "Free vibration analysis of the axially moving Levy-type viscoelastic plate", Eur. J. Mech. A-Soild., 29, 879-886.   DOI   ScienceOn
32 Mockensturm, E.M. and Guo, J. (2005), "Nonlinear vibration of parametrically excited, viscoelastic, axially moving strings", J. Appl. Mech., 72, 374-380.   DOI   ScienceOn
33 Nayfeh, A.H. (1993), Problems in Perturbation, Wiley, New York, USA.
34 Nayfeh, A.H. and Mook, D.T. (1979), Nonlinear Oscillations, Wiley, New York, USA.
35 Ozkaya, E., Pakdemirli, M. and Oz, H.R. (1997), "Non-linear vibrations of a beam-mass system under different boundary conditions", J. Sound. Vib., 199, 679-696.   DOI   ScienceOn
36 Pakdemirli, M. and Boyaci, H. (2003), "Non-linear vibrations of a simple-simple beam with a nonideal support in between", J. Sound. Vib., 268, 331-341.   DOI   ScienceOn
37 Pakdemirli, M. and Nayfeh, A.H. (1994), "Nonlinear Vibrations of a Beam-Spring-mass System", ASME J. Vib. Acoust., 116, 433-439.   DOI
38 Thomsen, J.J. (2003), Vibrations and Stability, Advanced Theory, Analysis, and Tools, Springer-Verlag, Berlin, Heidelberg.
39 Srinivasan, A. V. (1965), "Large amplitude-free oscillations of beams and plates", AIAA J., 3, 1951-1953.   DOI
40 Szemplinska-Stupnicka, W. (1990), The behaviour of nonlinear vibration systems, II., Kluwer, Netherlands.
41 Tseng, W.Y. and Dugundji, J. (1971), "Nonlinear vibrations of a buckled beam under harmonic excitation", ASME J. Appl. Mech., 38, 467-472.   DOI
42 Wrenn, B.G. and Mayers, J. (1970), "Nonlinear beam vibration with variable axial boundary restraint", AIAA J., 8, 1718-1720.   DOI
43 Zhang, N.H. and Chen, L.Q. (2005), "Nonlinear dynamical analysis of axially moving viscoelastic string", Chaos Soliton. Fract., 24(4),1065-1074.   DOI   ScienceOn
44 Zhang, N.H. (2008), "Dynamic analysis of an axially moving viscoelastic string by the Galerkin method using translating string eigenfunctions", Chaos Soliton. Fract., 35, 291-302.   DOI   ScienceOn